SUSY transformations with complex factorization constants. Application to spectral singularities

Abstract

Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero binorm (in the sense of distributions) at the singular point.

Thanks to A. Sokolov a number of inaccuracies are corrected

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• Spectral Singularities in the TE and TM modes of a PT-Symmetric Slab System: Optimal conditions for realizing a CPA-Laser
• Various Scattering Properties of a New PT-symmetric non-Hermitian potential
• Talbot self-imaging in $\mathcal{P}\mathcal{T}$-symmetric complex crystals
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